View
4
Download
0
Category
Preview:
Citation preview
1
Coherence Spaces for Resource-Sensitive Computation in Analysis
Kei Matsumoto(RIMS, Kyoto University)
2
Background● Computable analysis studies computation over topological spaces, by giving representations.
– Type two theory of Effectivity– Domain representations
● Their approaches are to track computation by continuous maps over “symbolic” spaces.
The principle: Computable ⇒ ContinuousBaire sp., Scott domains, ...
3
Our Proposal
● Our principle: Computable ⇒ Stable [Berry '78]Using instead of Scott-domains coherence spaces [Girard '86]. ● BTW, two morphisms coexists in coherence spaces:
stable & linear maps. ● A new question then arrises:
YX
FX Y
fTopological sp.
Coherence sp.
Tracked by stable map.
What are Linear Computations in Topology?
Girard's Linear LogicOne of the most influential papers in 80's in both logic and computer science.
● Restructuring both Classical & Intuitionistic Logic● Proof Nets● Resource-Conciousness
Attractive Ideas:
Girard's Linear Logic
● Restructuring both Classical & Intuitionistic Logic● Proof Nets● Resource-Conciousness
Attractive Ideas:
One of the most influential papers in 80's in both logic and computer science.
Girard's Linear Logic
● Restructuring both Classical & Intuitionistic Logic● Proof Nets● Resource-Conciousness
Attractive Ideas:
One of the most influential papers in 80's in both logic and computer science.
Girard's Linear Logic
● Restructuring both Classical & Intuitionistic Logic● Proof Nets● Resource-Conciousness
Attractive Ideas:
One of the most influential papers in 80's in both logic and computer science.
8
Resource-Sensitivity
Imagine:
there's no resource-conciousness
It isn't easy to do
Nothing to comsume or lose for
And no modalities too
Imagine all the people
Living life in Intuitionistic Logic ...
’’
9
Resource-Sensitivity
Ex. In Intuitionistic Logic, is true.
a person in the Intuitionistic Logic world
Substitute:● A:= “to pay ¥400”● B:= “to get a pack of cigarettes”● C:= “to get a cup of cake”
10
Resource-Sensitivity
Ex. In Intuitionistic Logic, is true.
Substitute:● A:= “to pay ¥400”● B:= “to get a pack of cigarettes”● C:= “to get a cup of cake”
11
Resource-Sensitivity
Ex. In Intuitionistic Logic, is true.
Substitute:● A:= “to pay ¥400”● B:= “to get a pack of cigarettes”● C:= “to get a cup of cake”
12
Resource-Sensitivity
Ex. In Intuitionistic Logic, is true.
Substitute:● A:= “to pay ¥400”● B:= “to get a pack of cigarettes”● C:= “to get a cup of cake”
Paradox
13
Resource-Sensitivity
Ex. In Intuitionistic Logic, is true.
Substitute:● A:= “to pay ¥400”● B:= “to get a pack of cigarettes”● C:= “to get a cup of cake”
Lack of conciousness to comsume assumptions !
A is used twice.
14
Resource-Sensitivity
Ex. In Linear Logic, is false. Because: In LL, we must use the assumption exactly once in the proof. Coherence Spaces are proposed as a denotational semantics which reflects this property. Via the Curry-Howard isomorphism, they are also a model of resource-sensitive computations of linear function programs.
New conjunction/ implication
15
Main Result from CCA’15Representations based on coherence spaces have an interesting feature: for every real funcitons, we have shown that
● stably realizable ⇔ continuous ● linearly realizable ⇔ uniformly continuous.
Let us emphasize that these correspondences hold for real functions.Next step: generalize them to a wider class.
16
Ⅰ. Review: Coherent Spaces
Ⅱ. Coherence as Uniformity
Ⅲ. Linear Admissibility
Ⅳ. Concluding Comments
17
Coherence Spaces
Def. A coherence space is a reflexive graph: ● a countable set of tokens with● a symmetric reflexive. binary rel. on
Write iff and (strict coherence)A clique is a set of tokens which are pairwise coherent. An anticlique is a set of tokens in which every pair is not coherent. ● :the set of all cliuqes. ● :the set of all finite cliques. ● :the set of all maximal cliques.
18
Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
19
Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
the “stage”“spotlights”
classified by “colors”
compact intervals are “projected”
20
Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
the “stage”“spotlights”
classified by “colors”
compact intervals are “projected”
21
Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
the “stage”“spotlights”
classified by “colors”
compact intervals are “projected”
22
Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
the “stage”“spotlights”
classified by “colors”
Lights of different colors are overlapped
23
Example: Cauchy SequencesLet . Each member of is identified with the dyadic rational as .For each , define
●
●
●
Ex. Define a coherence space
for dyadic Cauchy sequences as:
Maximal cliques ≈ (rapidly converging) Cauchy sequencesRealization of Real Numbers
24
Coherence as TopologyThe set of cliques is ordered by ⊆ , endowed with the Scott topology generated by the upper sets of finite cliques:
● Coherence spaces are very simplified domains● Compact Elements = finite cliques
Finite Cliques induce Topology
25
Stable & Linear MapsDef. A function is stable if it is -⊆ monotone and satisfies
Model of computations in which the amount of resources to be used is uniquely determined. Def. A function is linear if it is -⊆ monotone and satisfies
Model of computations in which resources are used exactly once. Stable & Linear Maps are Resource-Sensitive.
Collection of resources
26
Girard’s FormulaTwo closed structures of coherence spaces:
● The category Stbl of coh. spaces and stable maps is cartesian closed – : the coherence space for stable maps.
● The category Lin of coh. spaces and linear maps is monoidal closed – : the coherence space for linear maps.
They are combined by introducing the “of course” modality: naturally defined by .
Model of Intuit. Logic
Th.
Linear Decompostion of Cartesian closed Structure.
27
Ⅰ. Review: Coherent Spaces
Ⅱ. Coherence as Uniformity
Ⅲ. Linear Admissibility
Ⅳ. Concluding Comments
28
Review: Uniform Space
Uniform Covers are given horizontally
● A uniform space is a set with a uniformity: a collection of coverings of the set. – Each uniform cover is considered to be consisting of balls of the “same size”– They are partially ordered by the refinement relation and form a filter.
● They also induce a topology in the “vertical” way. – The balls surrounding each point form a neighborhood filter, which generates the uniform topology.
● Every uniformizable space has the finest uniformity: the fine uniform space.
29
Review: Uniform Space
Uniform Covers are given horizontally
● A uniform space is a set with a uniformity: a collection of coverings of the set. – Each uniform cover is considered to be consisting of balls of the “same size”– They are partially ordered by the refinement relation and form a filter.
● They also induce a topology in the “vertical” way. – The balls surrounding each point form a neighborhood filter, which generates the uniform topology.
● Every uniformizable space has the finest uniformity: the fine uniform space.
30
Review: Uniform Space
Uniform Covers are given horizontally
● A uniform space is a set with a uniformity: a collection of coverings of the set. – Each uniform cover is considered to be consisting of balls of the “same size”– They are partially ordered by the refinement relation and form a filter.
● They also induce a topology in the “vertical” way. – The balls surrounding each point form a neighborhood filter, which generates the uniform topology.
● Every uniformizable space has the finest uniformity: the fine uniform space.
31
Review: Uniform Space
neighbourhood filter
Neighbors are given vertically
● A uniform space is a set with a uniformity: a collection of coverings of the set. – Each uniform cover is considered to be consisting of balls of the “same size”– They are partially ordered by the refinement relation and form a filter.
● They also induce a topology in the “vertical” way. – The balls surrounding each point form a neighborhood filter, which generates the uniform topology.
● Every uniformizable space has the finest uniformity: the fine uniform space.
32
Review: Uniform Space
Th. Let be the fine space compatible with a topological space .
We’ve seen that situation!
● A uniform space is a set with a uniformity: a collection of coverings of the set. – Each uniform cover is considered to be consisting of balls of the “same size”– They are partially ordered by the refinement relation and form a filter.
● They also induce a topology in the “vertical” way. – The balls surrounding each point form a neighborhood filter, which generates the uniform topology.
● Every uniformizable space has the finest uniformity: the fine uniform space.
33
Girard’s FormulaTwo closed structures of coherence spaces:
● The category Stbl of coh. spaces and stable maps is cartesian closed – : the coherence space for stable maps.
● The category Lin of coh. spaces and linear maps is monoidal closed – : the coherence space for linear maps.
They are combined by introducing the “of course” modality: naturally defined by .
Model of Intuit. Logic
Th.
Linear Decompostion of Cartesian closed Structure.
34
Coherence as UniformityRecall: Incoherence implies disjointness: A partition of is an anticlique which induces the disjoint covering. Condition: a coherence space is disjointly coverable if every token can be extended to a partition:
Anticliques induce Uniformity
Th. 1) Partitions of form a subbasis for a uniformity. 2) Partitions of form a basis for a uniformity. 3) Both uniformities induce the Scott topology as the uniform topologies.4) The induced uniformity on is fine.
Prop. (homeomorphic.)
Coh. Sp. for anticliques
35
Cauchy Sequences AgainEx. Define a coherence space
for dyadic Cauchy sequences as:
• In any partition, spotlights of different colors must project disjoint sets by the second condition of the incoherence. • This is impossible essentially due to Sierpiński's theorem. • The partitions then generate the uniformity compatible with the real line, the representation is a uniformly open map.
Each is a partition of , because a maximal clique must contain “all colors”. There are no other partitions consisting of “several colors”.
( incoherence)
36
Linear ⇒ Uniform ContinuousRecall that
Th. Every linear map is uniformly continuous with respect to the uniformities induced by partitions. Cor. Every stable map is continuous.
(although it is a reinvention of the wheel...)
Th. Th.
We then combine these results.Assume that preserves maximality of cliques.
37
Ⅰ. Review: Coherent Spaces
Ⅱ. Coherence as Uniformity
Ⅲ. Linear Admissibility
Ⅳ. Concluding Comments
38
Standard RepresentationsLet be a Hausdorff uniform space with a countable basis consisting of countable coverings. Fact. Such a space is known to be separable metrizable. Def. The standard representation of is given by the coherence space defined by and
Each Maximal clique of specifies at most one point.
39
Linear AdmissibilityStandard representation does not depend on the choice of basis :This generalizes to the notion of admissibility. An Idea. A representation is linear admissible if 1. for every uniform cover of there exists a uniform cover of which refines , and2. for every subspace and its representation satisfying (1) the inclusion map is tracked by some linear map .
uniform cont.
A naive idea is to mimic Admissibility in TTE, but it doesn’t work!!
40
Linear AdmissibilityStandard representation does not depend on the choice of basis :This generalizes to the notion of admissibility. An Idea. A representation is linear admissible if 1. for every uniform cover of there exists a uniform cover of which refines , and2. for every subspace and its representation satisfying (1) the inclusion map is tracked by some linear map .
uniform cont.
A naive idea is to mimic Admissibility in TTE, but it doesn’t work!!
41
Linear AdmissibilityStandard representation does not depend on the choice of basis :This generalizes to the notion of admissibility. Def. A representation is linear admissible if 1. for every uniform cover of there exists a partition of which refines , and2. for every subspace and its representation satisfying (1) the inclusion map is tracked by some linear map .
Th. (1) Every uniformly continuous maps is then linearly realizable w.r.t. linear admissible representations.(2) Every standard representation is linear admissible.
strongly uniform cont.
42
Chain-Connectedness
Def. A uniform space is chain-connected if
A typical example: (though it is totally disconnected)In particular, every two members of a uniform cover is “chainly connected”
43
Chain-Connectedness
Def. A uniform space is chain-connected if
A typical example: (though it is totally disconnected)In particular, every two members of a uniform cover is “chainly connected”
44
Chain-Connectedness
Def. A uniform space is chain-connected if
A typical example: (though it is totally disconnected)In particular, every two members of a uniform cover is “chainly connected”
45
Chain-Connectedness
Def. A uniform space is chain-connected if
A typical example: (though it is totally disconnected)In particular, every two members of a uniform cover is “chainly connected”
46
Principal LemmaSuppose that is a chain-connected separable metrizable space and that is a uniform basis consiting of open coverings. N.B. Every uniform space has a basis of open coverings.Lem. The standard representation is a topologically and uniformly open map. ● Topological openness is immediate from the assumption.● Uniform openness is essentially due to “one-coloredness” of partitions.
47
Principal LemmaSuppose that is a chain-connected separable metrizable space and that is a uniform basis consiting of open coverings. N.B. Every uniform space has a basis of open coverings.Lem. The standard representation is a topologically and uniformly open map. ● Topological openness is immediate from the assumption.● Uniform openness is essentially due to “one-coloredness” of partitions.
48
Principal LemmaSuppose that is a chain-connected separable metrizable space and that is a uniform basis consiting of open coverings. N.B. Every uniform space has a basis of open coverings.Lem. The standard representation is a topologically and uniformly open map. ● Topological openness is immediate from the assumption.● Uniform openness is essentially due to “one-coloredness” of partitions.
49
Principal LemmaSuppose that is a chain-connected separable metrizable space and that is a uniform basis consiting of open coverings. N.B. Every uniform space has a basis of open coverings.Lem. The standard representation is a topologically and uniformly open map. ● Topological openness is immediate from the assumption.● Uniform openness is essentially due to “one-coloredness” of partitions.
50
Principal LemmaSuppose that is a chain-connected separable metrizable space and that is a uniform basis consiting of open coverings. N.B. Every uniform space has a basis of open coverings.Lem. The standard representation is a topologically and uniformly open map. ● Topological openness is immediate from the assumption.● Uniform openness is essentially due to “one-coloredness” of partitions.
51
Main Result
Th. If is chain-connected, ● : stably realizable ⇔ it is continuous.● : linearly realizable ⇔ it is uniformly continuous.Cor. If is linearly realizable then it is uniformly continuous on each chain-connected component.
Let and be separable metrizable uniform spaces with linear admissible representations, and .
Conversely, every uniformly continuous function is linearly realizable. To complete the corollary, we need to reduce the components by identifying some of them.
Recall: every separable metrizable space has the standard representation, hence has linear admissible representations.
52
Ⅰ. Review: Coherent Spaces
Ⅱ. Coherence as Uniformity
Ⅲ. Linear Admissibility
Ⅳ. Concluding Comments
53
Towards Linear Realizability● A linear combinatory algebra (LCA) U lin is defined from the “universal coherence space”. ● Coherent Representations = Modest Sets over Ulin● Mod (Ulin) is a model of linear logic. ● I'm essentially a realist...but still a bit a dreamer
● so I imagine there's a kind of “Linear Analysis” as the decomposition of Computable Analysis. – An analogy of the discovery of Linear Logic. – Every mathematical space has “admissible” representations in some sense, and functions are all linearly computable...
54
Towards Complexity in Analysis● It seems hopeless that the category of l inear admissible representations is monoidal closed because function spaces are not separable metrizable.● Nonetheless, I still believe that linearity is strongly related to uniform structures because:
Th (Férée-Gomaa-Hoyrup' 13). For any real functional
linear in our terminology
F is “uniformly continuous” w.r.t. a kind of uniformity on C[0,1]: We can explain this phenomenon in our model: Uniformity on C[0,1] a point in [0,1] & a uniformity of R
55
A Possible Way: Quasi-Uniformity● In some sense, separable metrisability of linear admissible representations seems inevitable:
– Because of countability of coherence spaces. ● We must have a countable basis of countable coverings.
● But... the Hausdorff property seems not necessary. ● just used for well-definedness of the representation.
● The use of non-Hausdorff metric seems a possible answer.● Every second-countable T1-space is separable quasi-metrizable.
– It is very likely that they have the standard representations for quasi-uniform spaces.
Recommended